Tensor products of spherical and
equivariant immersions
F. Decruyenaere∗
F. Dillen †
I. Mihai‡
L. Verstraelen
In [C1, C2, C3, C4], B.-Y. Chen introduced the tensor product of two immersions
of a given Riemannian manifold; he proved that the set of all immersions of the
given manifold, provided with direct sum and tensor product, defines a commutative
semiring.
In [DDVV] we introduced I, the commutative semiring of all transversal immersions of all differentiable manifolds in Euclidean spaces, provided with the binary
operations direct sum and tensor product. In this paper we further investigate which
immersions define a subsemiring or a multiplicative subsemigroup ; in particular, we
fix our attention on spherical immersions of differentiable manifolds, isometric and
equivariant immersions of Riemannian manifolds and immersions of finite type.
Denote by En the n-dimensional Euclidean space with Euclidean metric h , i.
The n-dimensional sphere with radius r is denoted by S n (r). Let f : M → Em be
an immersion of a differentiable manifold in a Euclidean space. Then f is said to be
transversal in a point p ∈ M if and only if the position vector f(p) is not tangent to
M at p, i.e. f(p) ∈
/ f∗ (TpM). If f is transversal in every point of M, then f shortly
is called transversal. Consider two differentiable manifolds M and N of dimensions
r resp. s and assume that f : M → Em and h : N → En are two transversal
immersions. Then the direct sum map f ⊕ h : M × N → Em+n : (p, q) 7→ (f(p), h(q))
and the tensor product map f ⊗ h : M × N → Emn : (p, q) 7→ f(p) ⊗ h(q) are
again two transversal immersions. We define a symmetric relation ∼ as follows : if
f : M → Em is an immersion and i : Em ⊂ En is a linear isometric immersion, then
∗
Supported by a research fellowship of the Research Council of the Katholieke Universiteit
Leuven
†
Senior Research Assistant of the National Fund for Scientific Research (Belgium)
‡
Research Fellow of the Research Council of the K.U.Leuven
Received by the editors November 1993
Communicated by A. Warrinier
AMS Mathematics Subject Classification : 53C40, 53B25, 58G25
Bull. Belg. Math. Soc. 1 (1994),643–648
644
F. Decruyenaere – F. Dillen – I. Mihai – L. Verstraelen
f ∼ i(f); if f : M → Em is an immersion and d : N → M is a diffeomorphism, then
f ∼ f ◦ d. Then ∼ is an equivalence relation on the set of all transversal immersions
Z and we put I = Z/ ∼. It is proved in [DDVV] that I, ⊕, ⊗ is a commutative
semiring, with e : {0} → E1 : 0 7→ 1 as identity element.
Theorem 1.
• The set of all spherical immersions of all differentiable manifolds induces a
subsemiring I∫ of I, ⊕, ⊗;
• the tensor product of two spherical immersions f and h of Riemannian manifolds M and N is an isometric immersion of the Riemannian product M × N
if and only if f(M) is contained in a sphere of radius r, h(N) is contained in
a sphere of radius r0 , f is a homothetical immersion with factor 1/r0 and h is
a homothetical immersion with factor 1/r
• the tensor product of two spherical isometric immersions f and h of Riemannian manifolds M and N such that both f(M) and h(N) are contained in
a sphere of radius 1, is an isometric immersion of the Riemannian product
M ×N
• if f and g are two homothetical spherical immersions in a sphere of radius 1
with the same factor λ, then f ⊗ g is again a homothetical immersion of the
Riemannian product with factor λ.
Corollary. The set of all spherical immersions of all differentiable manifolds
into a unit sphere induces a subsemigroup Iu∫ , ⊗ of I, ⊗.
Proof Let f : M → Em and h : N → En be immersions such that f(M) ⊆
S m−1 (r) and h(N) ⊆ S n−1 (s). Then, obviously, f and h are transversal immersions
such that f ⊗ h is also an immersion. It easily follows that
√
(f ⊕ h)(M × N) ⊆ S m+n−1 ( r2 + s2 )
and that
(f ⊗ h)(M × N) ⊆ S mn−1 (rs).
In order to prove (2), we assume that (M, g) and (N, g 0 ) are Riemannian manifolds. Take p ∈ M with vp , vp0 ∈ TpM and q ∈ N with wq , wq0 ∈ Tq N. Then
h(f ⊗ h)∗ (vp, wq ), (f ⊗ h)∗ (vp0 , wq0 )i = hf(p) ⊗ h∗ (wq ) + f∗ (vp) ⊗ h(q),
f(p) ⊗ h∗ (wq0 ) + f∗ (vp0 ) ⊗ h(q)i
=hf(p) ⊗ h∗ (wq ), f(p) ⊗ h∗ (wq0 )i + hf(p) ⊗ h∗ (wq ), f∗ (vp0 ) ⊗ h(q)i
(*)
+ hf∗ (vp ) ⊗ h(q), f(p) ⊗ h∗(wq0 )i + hf∗ (vp) ⊗ h(q), f∗(vp0 ) ⊗ h(q)i
=hf(p), f(p)ihh∗ (wq ), h∗ (wq0 )i + hf(p), f∗ (vp0 )ihh∗ (wq ), h(q)i+
hf∗ (vp ), f(p)ihh(q), h∗ (wq0 )i + hf∗ (vp ), f∗(vp0 )ihh(q), h(q)i
If hf(p), f(p)i = r2 and hh(q), h(q)i = r0 2 , then
h(f ⊗ h)∗ (vp, wq ), (f ⊗ h)∗ (vp0 , wq0 )i =hf∗ (vp), f∗ (vp0 )ir0 + hh∗ (wq ), h∗(wq0 )ir2
2
=g(vp, vp0 ) + g 0 (wq , wq0 ),
Tensor products of spherical and equivariant immersions
645
and thus f ⊗ h is an isometric immersions. The converse can be proved in the
same way, based on (∗). (3) is an immediate consequence and (4) can be proved
similarly as (2). 2
If (M, g) is a compact Riemannian homogeneous manifold, denote by G the
identity component of the group of all isometries of M. Then G is a compact Lie
group which acts transitively on M. Thus M = G/K, where K is the isotropy
subgroup of G at a point x ∈ M.
Definition. An (isometric) immersion f : (M, g) → Em is said to be equivariant
if and only if there exists a Lie homomorphism ψ : G → SO(m) such that f(q(p)) =
ψ(q)(f(p)), ∀q ∈ G and p ∈ M.
Theorem 2. The set E of all isometric equivariant transversal immersions of
all compact homogeneous Riemannian manifolds induces a subsemiring of I, ⊕, ⊗.
Proof. Let f : (M, g) → Em and h : (M 0 , g 0 ) → Em0 two isometric equivariant
transversal immersions of the compact homogeneous Riemannian manifolds (M, g)
and (M 0 , g 0). Denote by G (resp. G0 ) the identity component of the group of all
isometries of M (resp. M 0 ).
Then there exists ψ : G → SO(m) (resp. ψ 0 : G0 → SO(m0 )) such that
(
Define
by
f(q(α)) = ψ(q)(f(α)),
∀α ∈ M, q ∈ G
0
0 0
h(q (β)) = ψ (q )(h(β)), ∀β ∈ M 0 , q 0 ∈ G0 .
(
(
ψ⊕ : G × G0 → SO(m + m0 )
ψ⊗ : G × G0 → SO(mm0 )
ψ⊕ (q, q 0) = ( ψ(q) 00 ψ 0(q 0) )
ψ⊗ (q, q 0) = ψ(q) ⊗ ψ(q 0).
Then we have
(f ⊕ h)((q, q 0)(α, β)) = (f(q(α)), h(q 0(β)))
= (ψ(q)(f(α)), ψ 0(q 0)(h(β))
= ψ⊕ (q, q 0)(f(α), h(β))
= ψ⊕ (q, q 0)[(f ⊕ h)(α, β)]
(f ⊗ h)((q, q 0)(α, β)) = f(q(α)) ⊗ h(q 0(β))
= ψ(q)(f(α)) ⊗ ψ 0(q 0)(h(β))
= ψ⊗ (q, q 0)[(f ⊗ h)(α, β)]
Thus, f ⊕ h and f ⊗ h are isometric equivariant immersions. 2
We now concentrate on the tensor product of spherical isometric immersions
(with spherical it is meant that the image is contained in a sphere of radius 1).
646
F. Decruyenaere – F. Dillen – I. Mihai – L. Verstraelen
Lemma A.Assume that f : (M, g) → Em and h : (N, g 0 ) → En are two spherical
isometric immersions with Im(f) ⊆ S m−1 (R) and Im(h) ⊆ S n−1 (R0 ). Let p ∈ M
and q ∈ N. Suppose dim M = r with basis of Tp M given by e1, . . . , er and let a
basis of the normal bundle of f(M) in S m−1 (R) at p be given by ξ1 , ξ2, . . . , ξm−r−1 ;
suppose dim N = s with basis of Tq N given by f1, . . . , fs and let a basis of the
normal bundle of h(N) in S n−1 (R0 ) at q be given by ζ1 , ζ2 , . . . , ζn−s−1 . Then a basis
of (f ⊗ h)∗(T(p,q)(M × N)) is given by
{f(p) ⊗ h∗ (fj ), f∗(ei ) ⊗ h(q)|1 ≤ i ≤ r, 1 ≤ j ≤ s}.
A basis of the normal space of (f ⊗ h)(M × N) in S mn−1 (RR0 ) at (p, q) is given by
{f∗ (ei) ⊗ h∗ (fj ), 1 ≤ i ≤ r, 1 ≤ j ≤ s;
f∗(ei ) ⊗ ζj
, 1 ≤ i ≤ r, 1 ≤ j ≤ n − s − 1;
ξi ⊗ h∗ (fj )
ξi ⊗ ζj
, 1 ≤ i ≤ m − r − 1, 1 ≤ j ≤ s;
, 1 ≤ i ≤ m − r − 1, 1 ≤ j ≤ n − s − 1;
ξi ⊗ h(q)
, 1 ≤ i ≤ m − r − 1;
f(p) ⊗ ζj
, 1 ≤ j ≤ n − s − 1}.
Proof. The first part follows from [DDVV], Lemma 2. The second part is then
obvious. 2
Lemma B.With the same notations as in Lemma A, put R = R0 = 1, X =
(X1 , X2 ), Y = (Y1 , Y2 ) ∈ Tp,q (M × N). If the second fundamental forms of f : M →
S m−1 (1) and h : N → S n−1 (1) are given by σf and σh , then the second fundamental
form of f ⊗ h : M × N → S mn−1 is given by
σf ⊗h (X, Y ) = f∗ (X1 )⊗h∗ (Y2 )+f∗ (Y1 )⊗h∗ (X2 )+f(p)⊗σh (X2 , Y2 )+σf (X1 , Y1 )⊗h(q).
Proof. Let D denote the affine connection of Euclidean space, and let ∇1 and
∇ denote the Levi Civita connections of M and N. Then
2
DX (f ⊗ h)∗ (Y ) = DX (f(p) ⊗ h∗ (Y2 )) + DX (f∗(Y1 ) ⊗ h(q))
= DX1 f(p) ⊗ h∗ (Y2 ) + f(p) ⊗ DX2 h∗ (Y2 )
+ DX1 f∗ (Y1 ) ⊗ h(q) + f∗ (Y1 ) ⊗ DX2 h(q)
= f∗ (X1 ) ⊗ h∗(Y2 ) + f(p) ⊗ h∗ (∇2X2 Y2 ) + f(p) ⊗ σh (X2 , Y2 )
− hY1 , Y2if(p) ⊗ h(q) + f∗ (∇1X1 Y1 ) ⊗ h(q) + σf (X1 , Y1) ⊗ h(q)
− hX1 , X2 if(p) ⊗ h(q) + f∗ (Y1 ) ⊗ h∗ (X2 ).
The result now follows from Lemma A.
In the following theorem we use the same notations as in Lemma A.
Tensor products of spherical and equivariant immersions
647
Theorem 3.
• The set Imu∫ of all minimal spherical isometric immersions of all Riemannian
manifolds into a unit sphere is a subsemigroup of Iu∫ , ⊗;
• The set I{tu∫ of all finite type spherical isometric mmersions of all Riemannian
manifolds into a unit sphere is a subsemigroup of Iu∫ , ⊗;
• The set It∇∫ of all spherical isometric immersions of all manifolds into a
sphere S 2m−1 which are totally real with respect to at least one complex structure on E2m is an ideal of I∫ , ⊕, ⊗;
Proof. Assume f and h are minimal spherical isometric immersions; then it is
easy to see that the mean curvature vector of the tensor product immersion
Hf ⊗h (p, q) =
s
r
X
1 X
( σf ⊗h ((0, fj ), (0, fj )) +
σf ⊗h ((ei , 0), (ei , 0))).
r + s j=1
i=1
Therefore, by Lemma B,
Hf ⊗h (p, q) =
s
r
X
1 X
σf (ei , ei) ⊗ h(q))
( f(p) ⊗ σh (fj , fj ) +
r + s j=1
i=1
and thus
Hf ⊗h =
1
(sf ⊗ Hh + rHf ⊗ h).
r+s
It is now clear that if f and h are minimal, then f ⊗ h is minimal too.
In order to prove (2), assume that f = f1 + . . . + ft with ∆f fi = λi fi (1 ≤ i ≤ t)
P
and h = h1 + . . . + hu with ∆h hj = µj hj (1 ≤ j ≤ u). Then f ⊗ h = i,j fi ⊗ hj with
∆(fi ⊗ hj ) = (λi + µj )(fi ⊗ hj ) and thus f ⊗ h is of finite type ≤ tu.
For proving (3), we first remark that the direct sum of any two totally real
immersions is again totally real. Next let f : M → S 2m−1 be a totally real spherical
immersion with respect to a complex structure J on E2m , and let h : N → S n−1
be a spherical immersion. We prove that f ⊗ h is totally real with respect to the
complex structure J ⊗ I on E2mn . This follows immediately from
h(J ⊗ I)(f ⊗ h)∗ (vp, wq ), (f ⊗ h)∗ (vp0 , wq0 )i
=hJ f(p), f(p)ihh∗ (wq ), h∗(wq0 )i + hJ f(p), f∗ (vp0 )ihh∗ (wq ), h(q)i
+hJ f∗ (vp), f(p)ihh(q), h∗ (wq0 )i + hJ f∗(vp ), f∗(vp0 )ihh(q), h(q)i = 0.2
Examples. The tensor product of two circles of radius 1 is a flat surface of type
1 in a 3-dimensional unit sphere. In particular, it is the Clifford torus. The tensor
product of a circle of radius 1 and a small circle on a 2-dimensional unit sphere is
a flat surface of type 2. The tensor product of two small circles on a 2-dimensional
unit sphere is a flat surface of type 3. In general, the tensor product of two curves
of finite type on a unit sphere (cf. [CDDVV]) is a flat torus of finite type.
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F. Decruyenaere – F. Dillen – I. Mihai – L. Verstraelen
References
[C1]
B.-Y. Chen, Differential Geometry of semiring of immersions, I: General
Theory Bull. Inst. Math. Acad. Sinica21 (1993) 1–34
[C2]
B.-Y. Chen, Classification of tensor product immersions which are of 1-type
to appear in Glasgow J. Math
[C3]
B.-Y. Chen, Two theorems on tensor product immersions Rend. Sem. Mat.
Messina, Série II, Tomo XIV, Vol I (1991) 69—83
[C4]
B.-Y. Chen, Differential geometry of tensor product immersions Ann. Global
Anal. Geom. to appear
[CDDVV] B. Y. Chen, J. Deprez, F. Dillen, L. Verstraelen and L. Vrancken, Curves
of finite type, Geometry and Topology of Submanifolds II (1990) World
Scientific, Singapore 76–110
[DDVV] F. Decruyenaere, F. Dillen, L. Verstraelen, L. Vrancken, The semiring of
immersions of manifolds Beiträge zur Algebra und Geometrie 34 (1993)
209–215 to appear
F. Decruyenaere – F. Dillen – I. Mihai – L. Verstraelen
Katholieke Universiteit Leuven, Departement Wiskunde,
Celestijnenlaan 200B, B-3001 Leuven, Belgium
L. Verstraelen
Vrije Universiteit Brussel, Group of Exact Sciences,
Vrijheidslaan 17, B-1080 Brussel, Belgium